Matrix theory

Matrix theory is a branch of mathematics which focuses on the study of matrices. Initially, a sub-branch of linear algebra, it has grown to cover subjects related to graph theory, algebra, combinatorics and statistics as well.

Related Topics:
Mathematics - Matrices - Linear algebra - Graph theory - Algebra - Combinatorics - Statistics

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A matrix is a rectangular array of numbers. For an elementary article on matrices, their basic properties, and history, see the article matrix (mathematics).

Related Topics:
Array - Matrix (mathematics)

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A matrix can be identified with a linear transformation between two vector spaces. Therefore matrix theory is usually considered as a branch of linear algebra. The square matrices play a special role, because the n×n matrices for fixed n have many closure properties.

Related Topics:
Linear transformation - Vector space - Linear algebra - Square matrices - Closure

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In graph theory, each labeled graph corresponds to a unique non-negative matrix, the adjacency matrix. A permutation matrix is the matrix representation of a permutation; it is a square matrix with entries 0 and 1, with just one entry 1 in each row and each column. These types of matrices are used in combinatorics.

Related Topics:
Graph theory - Non-negative matrix - Adjacency matrix - Permutation matrix - Permutation - Combinatorics

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The ideas of stochastic matrix and doubly stochastic matrix are important tools to study stochastic processes, in statistics.

Related Topics:
Stochastic matrix - Stochastic process - Statistics

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Positive-definite matrices occur in the search for maxima and minima of real-valued functions, when there are several variables.

Related Topics:
Positive-definite - Maxima and minima

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It is also important to have a theory of matrices over arbitrary rings. In particular, matrices over polynomial rings are used in control theory.

Related Topics:
Ring - Polynomial ring - Control theory

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Within pure mathematics, matrix rings can provide a rich field of counterexamples for mathematical conjectures, amongst other uses.

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